Perpetuity Calculator
Present value of a level or growing perpetuity, with payments at the start or end of each period.
What it does
A perpetuity is a payment stream with no end date. Console bonds, preferred stock dividends, and the terminal value in a discounted cash flow model all use the same idea: a regular payment that keeps arriving forever. Because the stream never stops, its value does not grow without limit. Each distant payment is discounted so heavily that the total converges to a finite number.
For a level payment the arithmetic is short. Divide the payment by the discount rate. At a 5% discount rate, $10,000 a year forever is worth $200,000. The result is the amount that, invested at 5%, would produce $10,000 a year while leaving the principal intact.
A growing perpetuity allows the payment to rise at a steady rate. The denominator becomes r minus g, so the value rises as growth approaches the discount rate. With $10,000 next year, a 5% discount rate, and 2% growth, the value is $10,000 / (0.05 - 0.02) = $333,333.33. The growth assumption carries a lot of weight, which is why the formula is sensitive near the point where r and g meet.
Timing changes the answer by one period. If each payment arrives at the start of the period rather than the end, every cash flow shifts one period closer, and the whole present value is multiplied by 1 + r. This is the annuity-due convention. The calculator handles both cases with the same inputs.
The model assumes the payment and growth rate stay constant forever. Real assets rarely behave that way, so the result is best read as a terminal value or a first approximation. It is also undefined when growth equals or exceeds the discount rate, because the sum diverges.
How to use
- Enter the payment expected in the next period, in dollars.
- Enter the discount rate per period as a percentage.
- Enter the growth rate per period; use 0 for a level perpetuity.
- Choose whether payments arrive at the end or the start of each period.
- Read the present value, the value per dollar of payment, and the net discount rate.
- Check that the growth rate stays below the discount rate before relying on the result.
Formula
- PV
- Present value of the payment stream, in dollars.
- C
- Payment expected in the next period, in dollars.
- r
- Discount rate per period, written as a decimal (5% becomes 0.05).
- g
- Constant growth rate per period, written as a decimal (2% becomes 0.02).
- 1 + r
- Factor applied to the ordinary result when each payment comes at the start of the period.
With g = 0 the formula collapses to C / r. As g approaches r the denominator shrinks and the present value rises sharply, so small changes in either rate can move the result by a large amount. When g is greater than or equal to r the series does not converge and the calculator returns an error instead of a number. The growth rate should describe a nominal payment stream when the discount rate is nominal, and a real stream when the discount rate is real.
Worked example
Example: $10,000 next year, 5% discount rate, 2% growth
Payment of $10,000 next year, discount rate 5% per year, growth 2% per year, payments at the end of each year.
- Convert the rates to decimals: r = 0.05 and g = 0.02.
- The net discount rate is r - g = 0.05 - 0.02 = 0.03, or 3%.
- Divide the payment by the net rate: $10,000 / 0.03 = $333,333.33.
- Because payments arrive at the end of each year, no timing adjustment is needed.
- The value per dollar of payment is $333,333.33 / $10,000 = 33.33x.
- Check the first few terms: $9,523.81 + $9,070.29 + $8,638.38 converges toward $333,333.33.
Reference table
| Discount rate | Growth rate | Net rate | Value per $1 |
|---|---|---|---|
| 4% | 0% | 4% | $25.00 |
| 5% | 0% | 5% | $20.00 |
| 5% | 2% | 3% | $33.33 |
| 6% | 3% | 3% | $33.33 |
| 8% | 5% | 3% | $33.33 |
| 10% | 9% | 1% | $100.00 |
Limits
- Assumes a constant payment and a constant growth rate for the entire stream.
- Requires the discount rate to exceed the growth rate; otherwise the value is infinite.
- Ignores taxes, fees, default risk, and any change in the discount rate over time.
- A perpetuity with payments at the start of each period is an approximation of a stream that begins immediately.
Common mistakes
- Entering rates as whole numbers but treating them as decimals in a hand check.
- Using a growth rate above the discount rate and expecting a finite value.
- Discounting the first payment twice when the payment is already stated as next period's amount.
- Mixing a nominal discount rate with a real growth rate.
- Forgetting the 1 + r factor when payments arrive at the start of each period.
Try it yourself
Quick checks. Answers sit tucked under each question.
$5,000 next year, 6% discount rate, 1% growth, end of period?
Net rate is 5%, so the present value is $100,000.00, or 20.00x the payment.
$2,000 next year, 4% discount rate, no growth, start of period?
The ordinary value is $50,000.00, multiplied by 1.04 for start-of-period timing, for $52,000.00 and 26.00x.
$1,000 next year, 8% discount rate, 3% growth, end of period?
Net rate is 5%, so the present value is $20,000.00, or 20.00x the payment.
Questions
What is the difference between a perpetuity and an annuity?
An annuity ends after a set number of periods. A perpetuity keeps paying forever, which removes the period count from the formula.
Why does growth lower the denominator?
Each payment is larger than the one before, so the stream is worth more. Dividing by r - g captures that extra value in one step.
Can the growth rate be negative?
Yes in principle, for a declining stream. This calculator accepts zero or positive growth, which covers the common valuation case.
What happens when growth equals the discount rate?
The denominator becomes zero and the present value is infinite. The series does not converge, so no finite answer exists.
When do I use the start-of-period option?
Use it when the first payment is made immediately rather than one period from now, as with rent paid in advance.
Is a perpetuity value the same as terminal value?
It is one common way to estimate terminal value: the value at the end of a forecast horizon of all cash flows after that point.
Sources
- NYU Stern, Aswath Damodaran: present value and growing perpetuity formulas
- MIT OpenCourseWare 15.414: perpetuity and growing perpetuity review notes